Molecular potential coordinate system

The structure of this chapter is as follows. In Section II, after the concept of potential-energy surface and the coordinate systems in which the potential can be represented have been introduced, we describe the most important topographical characteristics of the molecular potential function. The general aspects, which refer to the calculation of the potential energy by ab initio methods, are analyzed in Section III. The need to develop efficient methods for the calculation of the potential function and the corresponding gradient in... 

Let us consider a nucleus with I > 1 in atom 1 of a diatomic molecule. We shall use a local coordinate system (x, y, z) with its origin at the nucleus and z lying along the molecular bond. An unscreened electric charge q at a distance r from the nucleus gives rise to an electrostatic potential... 

While the model employed in the present work provides a reasonable picture of a unimolecular reaction involving a large molecule in solution, other ingredients not considered here may play a role in some systems. The possible role played by intramolecular friction (nonlinear coupling between the reaction coordinate and other nonreactive modes near the barrier) has been discussed in Section IV. Also, the dependence of the molecular potential surface, in particular the activation barrier on the molecule-solvent interaction, may dominate in some cases the observed solvent effect on the rate. Such may be the case (see Section VIII) in a polar solvent when the reaction involves a change in the molecular dipole moment (such as a charge transfer reaction). 

In the laboratory frame the motion of the three particles depends on nine variables, three of which define the position of the center-of-mass. Other three coordinates are needed to describe the rotation of the system in the space and therefore the internal motion is described by the three remaining coordinates. For example, in molecular dynamics the potential energy surface in general is calculated and presented using geometrical coordinates, such the interparticle distances, or two bond distances and an angle. But it is convenient and necessary to use different coordinate systems to describe and understand the dynamics of the particles, because of the rotational terms which appear in the full Hamiltonian. In this context, we will present the transformation equations from the interparticle distances to coordinate sets of the hyperspherical and related types, successful in the treatment of the dynamics. 

The radial deformation of the valence density is accounted for by the expansion-contraction variables (k and k ). The ED parameters P, Pim , k, and k are optimized, along with conventional crystallographic variables (Ra and Ua for each atom), in an LS refinement against a set of measured structure factor amplitudes. The use of individual atomic coordinate systems provides a convenient way to constrain multipole populations according to chemical and local symmetries. Superposition of pseudoatoms (15) yields an efficient and relatively simple analytic representation of the molecular and crystalline ED. Density-related properties, such as electric moments electrostatic potential and energy, can readily be obtained from the pseudoatomic properties [53]. 

In general the Hartree-Fock equations for any molecular system form a set of 3-dimensional partial differential equations for orbitals, Coulomb and exchange potentials. In the case of diatomic molecules the prolate spheroidal coordinate system can be used to describe the positions of electrons and one of the coordinates (the azimuthal angle) can be treated analytically. As a result one is left with a problem of solving second order partial differential equations in the other two variables, (rj and ). 

---